What other countries describe is the equivalent of your public health system. If you're willing to pay out of pocket, like you describe, you can have a similar experience in the US. However, that can get expensive quickly since you normally don't have just one appointment with a vascular neurosurgeon.
But the thing is that paying out of pocket would be very expensive in the "developed" countries.
The advantage of the less developed countries is that you can get pretty good private medical care for a reasonably price. The quality may not match the very best care available in the rich countries, but in many cases, it is adequate.
You would think so, but insurance has infected everything.
I often try to go cash pay, since in many cases at this point it's actually cheaper, absurdly enough, and get told that I can't even get an appointment that way. I actually got kicked off primary care at a regional (fairly large) practice because they couldn't take my insurance, would not bill out of network, didn't service uninsured people (!!), and would not take cash, credit, barter or trade.